On the periodic Korteweg-de Vries equation: a normal form approach

This paper discusses an improved smoothing phenomena for low-regularity solutions of the Korteweg-de Vries (KdV) equation in the periodic settings by means of normal form transformation. As a result, the solution map from a ball on $H^{-1/2+}$ to $C_0^t ([0,T], H^{-1/2+})$ can be shown to be Lipschitz in a $H^{0+}_x$ topology, where the Lipschitz constant only depends on the rough norm $\|u_0\|_{H^{-1/2+}}$ of the initial data. A similar episode has been observed in a recent paper on 1D quadratic Schrödinger equation in low-regularity setting.

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